Functorial Knot Theory: Categories Of Tangles, Coherence, Categorical Deformations And Topological Invariants
Seiten
2001
World Scientific Publishing Co Pte Ltd (Verlag)
978-981-02-4443-9 (ISBN)
World Scientific Publishing Co Pte Ltd (Verlag)
978-981-02-4443-9 (ISBN)
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A study of functorial knot theory. It discusses the key ideas in the discovery of monoidal categories of tangles as central objects in low-dimensional topology. The focus then turns to the deformation theory of monoidal categories and the related deformation theory of monoidal functors.
Almost since the advent of skein-theoretic invariants of knots and links (the Jones, HOMFLY, and Kauffman polynomials), the important role of categories of tangles in the connection between low-dimensional topology and quantum-group theory has been recognized. The rich categorical structures naturally arising from the considerations of cobordisms have suggested functorial views of topological field theory.This book begins with a detailed exposition of the key ideas in the discovery of monoidal categories of tangles as central objects of study in low-dimensional topology. The focus then turns to the deformation theory of monoidal categories and the related deformation theory of monoidal functors, which is a proper generalization of Gerstenhaber's deformation theory of associative algebras. These serve as the building blocks for a deformation theory of braided monoidal categories which gives rise to sequences of Vassiliev invariants of framed links, and clarify their interrelations.
Almost since the advent of skein-theoretic invariants of knots and links (the Jones, HOMFLY, and Kauffman polynomials), the important role of categories of tangles in the connection between low-dimensional topology and quantum-group theory has been recognized. The rich categorical structures naturally arising from the considerations of cobordisms have suggested functorial views of topological field theory.This book begins with a detailed exposition of the key ideas in the discovery of monoidal categories of tangles as central objects of study in low-dimensional topology. The focus then turns to the deformation theory of monoidal categories and the related deformation theory of monoidal functors, which is a proper generalization of Gerstenhaber's deformation theory of associative algebras. These serve as the building blocks for a deformation theory of braided monoidal categories which gives rise to sequences of Vassiliev invariants of framed links, and clarify their interrelations.
Part 1 Knots and categories: monoidal categories, functors and natural transformations; a digression on algebras; knot polynomials; smooth tangles and PL tangles; a little enriched category theory. Part 2 Deformations: deformation complexes of semigroupal categories and functors; first order deformations; units; extrinsic deformations of monoidal categories; categorical deformations as proper generalizations of classical notions. (Part contents).
| Erscheint lt. Verlag | 18.4.2001 |
|---|---|
| Reihe/Serie | Series on Knots & Everything ; 26 |
| Verlagsort | Singapore |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie | |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-10 | 981-02-4443-6 / 9810244436 |
| ISBN-13 | 978-981-02-4443-9 / 9789810244439 |
| Zustand | Neuware |
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